Symmetry and Conservation Laws in Semiclassical Wave Packet Dynamics

TL;DR

This paper explores symmetries and conserved quantities in semiclassical Gaussian wave packet dynamics, revealing their symplectic structure and deriving conserved angular momentum through Noether's theorem.

Contribution

It introduces a unified framework for identifying symmetries and conservation laws in semiclassical wave packet dynamics, connecting classical, quantum, and geometric perspectives.

Findings

Semiclassical angular momentum is conserved and mirrors classical properties.

The dynamics possess a symplectic and Hamiltonian structure.

Natural symmetry groups lead to conserved quantities via momentum maps.

Abstract

We formulate symmetries in semiclassical Gaussian wave packet dynamics and find the corresponding conserved quantities, particularly the semiclassical angular momentum, via Noether's theorem. We consider two slightly different formulations of Gaussian wave packet dynamics; one is based on earlier works of Heller and Hagedorn, and the other based on the symplectic-geometric approach by Lubich and others. In either case, we reveal the symplectic and Hamiltonian nature of the dynamics and formulate natural symmetry group actions in the setting to derive the corresponding conserved quantities (momentum maps). The semiclassical angular momentum inherits the essential properties of the classical angular momentum as well as naturally corresponds to the quantum picture.